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arXiv · 2412.03034

Low-Lying Zeros of $L$-functions of Ad\'elic Hilbert Modular Forms and their Convolutions

Alia Hamieh·Peng-Jie Wong
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Abstract

In this article, we study the density conjecture of Katz and Sarnak for $L$-functions of ad\'elic Hilbert modular forms and their convolutions. In particular, under the generalised Riemann hypothesis, we establish several instances supporting the conjecture and extending the works of Iwaniec-Luo-Sarnak and many others. For applications, we obtain an upper bound for the average order of $L$-functions of Hilbert modular forms at $s=\frac{1}{2}$ as well as a positive proportion of non-vanishing of certain Rankin-Selberg $L$-functions.

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Alia Hamieh, Peng-Jie Wong. 2024-12-04. Low-Lying Zeros of $L$-functions of Ad\'elic Hilbert Modular Forms and their Convolutions. https://arxiv.org/abs/2412.03034

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